A Modified Generalized Analytic Feynman Integral Associated with the Bounded Linear Operator

Abstract

In this paper, we define a modified and generalized analytic Feynman integral associated with the bounded linear operator on abstract Wiener spaces. We then prove its existence. We also establish some modified and generalized analytic Feynman integration formulas and relationships involving the generalized Cameron–Storvick theorem. Finally, we give some examples to explain the usefulness of our research.

1. Introduction

Let H be a real separable infinite-dimensional Hilbert space with the norm ${|·|}_H=\sqrt{{\langle ·,·\rangle }_H}$, where ${\langle ·,·\rangle }_H$ denotes the inner product and let B the completion of H with respect to ${\parallel ·\parallel }_0$ where ${\parallel ·\parallel }_0$ is a measurable norm on H with respect to the Gaussian cylinder measure ${\nu }_0$ on H. Let $\mathbf{i}$ denote the natural injection from H into B. Then, the adjoint operator ${\mathbf{i}}^{*}$ of $\mathbf{i}$ is one-to-one and it maps $B^{*}$ continuously onto a dense subset, $H^{*}$, where $B^{*}$ and $H^{*}$ are topological duals of B and H, respectively. By identifying $H^{*}$ with H and $B^{*}$ with ${\mathbf{i}}^{*}{B}^{*}$, we have a triple $B^{*}\subset H^{*}\approx H\subset B$. By a well-known result of Gross [1,2], ${\nu }_0\circ {\mathbf{i}}^{-1}$ has a unique countably additive extension $\nu$ to the Borel $\sigma$-algebra $\mathcal{B}\left(B\right)$ of B. The triple $(B,H,\nu )$ is called an abstract Wiener space; for a more detailed study of the abstract Wiener space, see [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18].

Many research results on the analytic Feynman integral using Feynman’s idea have been studied and published in [6,7,19,20,21,22,23,24]. Feynman’s idea is to use his integral in a formal manner in his approach to quantum mechanics. One of famous methods that provides a rigorously meaningful definition of Feynman’s integral is based on the similarity between the Wiener and Feynman integrals, and several procedures were set up by many mathematicians to obtain Feynman integrals from Wiener integrals by an analytic continuation from the real axis to the imaginary axis.

In [6,22], the authors introduced the concept of a generalized analytic Feynman integral and pointed out its importance and the usefulness. Furthermore, they established various generalized analytic Feynman integration formulas. The concept of generalized analytic Feynman integral can be used to explain the observation that the status of the harmonic oscillator can be exchanged by the status of the anharmonic oscillator in certain physical circumstances by choosing the function.

In this paper, we introduce a more modified generalized analytic Feynman integral associated with the bounded linear operator on abstract Wiener spaces. We then obtain the existence and establish some modified generalized analytic Feynman integration formulas and relationships involving the Cameron–Strovick theorem. Finally, we provide some examples to illustrate the usefulness of the modified generalized analytic Feynman integrals. Depending upon the choice of the bounded linear operator, most research results in previous papers can be obtained as corollaries of our results.

2. Definitions and Preliminaries

In this section, we list some definitions, notations, and concepts used in this paper.

For an integrable functional F, the (abstract) Wiener integral of F is denoted by

$${\int }_{B}F\left(x\right)d\nu \left(x\right).$$

For each $h\in H$ and $x\in B$, we define a stochastic inner product ${(h,x)}^{\sim }$ by

$${(h,x)}^{\sim }=\left\{\begin{array}{cc}\underset{n\to \infty }{lim}\sum _{j=1}^n{\langle h,e_j\rangle }_H(e_j,x),\hfill & \mathrm{if}\mathrm{the}\mathrm{limit}\mathrm{exists},\hfill \\ 0,\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

where $(·,·)$ is the natural dual paring on $B^{*}\times B$, and ${\left\{e_j\right\}}_{j=1}^{\infty }$ is a complete orthonormal set in H with $e_j$ being in $B^{*}$ for all $j=1,2,\dots$. Then, we have the following assertions:

(i)

For each $h\ne 0$ in $H,$${(h,·)}^{\sim }$ exists for all $x\in B$.

(ii)

${(h,·)}^{\sim }$ is a Gaussian random variable on B with a mean of zero and variance ${\left|h\right|}^2$.

(iii)

${(h,x)}^{\sim }$ is essentially independent of the choice of the complete orthonormal set.

(iv)

We have an integration formula

$${\int }_{B}exp\left\{i{(h,x)}^{\sim }\right\}d\nu \left(x\right)=exp\left\{-\frac{1}{2}{\left|h\right|}_H^2\right\}.$$

(1)

For a more detailed study of the stochastic inner product ${(h,x)}^{\sim }$, see [4,5,10,11,25].

A subset E of an abstract Wiener product space B is said to be scale-invariant measurable provided that $\{\rho x:x\in E\}$ is abstract Wiener measurable for every $\rho$, and a scale-invariant measurable set N of B is said to be a scale-invariant null provided that $\nu \left(\right\{\rho x:x\in N\left\}\right)=0$ for any $\rho >0$. A property that holds except on a scale-invariant null set is said to hold as scale-invariant almost everywhere (s-a.e.). A functional F on B is said to be scale-invariant measurable provided that F is defined on a scale-invariant measurable set and $F(\rho ·)$ is measurable for any $\rho$. If two functionals F and G on B are equal s-a.e., i.e., for any $\rho$, $\nu \left(\right\{x\in B:F\left(\rho x\right)\ne G\left(\rho x\right)\left\}\right)=0$, then we say two functionals F and G coincide with s-a.e. with respect to $\nu$ [24].

We are ready to state the definition of the modified generalized analytic Feynman integral associated with the bounded linear operator.

Definition 1. 

Let $\mathbb{C}$ denote the complex numbers, let ${\mathbb{C}}_{+}=\{\lambda \in \mathbb{C}:\mathrm{Re}\left(\lambda \right)>0\}$, and let ${\tilde{\mathbb{C}}}_{+}=\{\lambda \in \mathbb{C}:\lambda \ne 0\mathit{and}\mathrm{Re}\left(\lambda \right)\ge 0\}$. Let S be a bounded linear operator on B, and let h be an element of H. Let $F:B⟶\mathbb{C}$ be a functional such that for each $\lambda >0$, the Wiener integral

$$J_S^{c_{\lambda },h}\left(F\right)\left(\lambda \right)\equiv {\int }_{B}F({\lambda }^{-\frac{1}{2}}Sx+c_{\lambda }h)d\nu \left(x\right)$$

exists, where $c_{\lambda }$ is a constant that depends on λ. If there exists a function ${\left(J_S^{c_{\lambda },h}\left(F\right)\right)}^{*}\left(\lambda \right)$ that is analytic in ${\mathbb{C}}_{+}$ such that ${\left(J_S^{c_{\lambda },h}\left(F\right)\right)}^{*}\left(\lambda \right)=J_S^{c_{\lambda },h}\left(F\right)\left(\lambda \right)$ for all $\lambda >0$, then ${\left(J_S^{c_{\lambda },h}\left(F\right)\right)}^{*}\left(\lambda \right)$ is defined to be the generalized analytic Wiener integral of F over B with parameters λ, and for $\lambda \in {\mathbb{C}}_{+}$, we write

$${\left(J_S^{c_{\lambda },h}\left(F\right)\right)}^{*}\left(\lambda \right)={\int }_B^{a{n}_{S,\lambda }^{c_{\lambda },h}}F\left(x\right)d\nu \left(x\right).$$

Let q be a nonzero real number and let F be a functional such that ${\left(J_S^{c_{\lambda },h}\left(F\right)\right)}^{*}\left(\lambda \right)$ exists for all $\lambda \in {\mathbb{C}}_{+}$. If the following limit exists, we call it the modified generalized analytic Feynman integral of F with parameter q, and we write

$${\int }_B^{an{f}_{S,q}^{c_q,h}}F\left(x\right)d\nu \left(x\right)=\underset{\begin{array}{c}\lambda \to -iq\end{array}}{lim}{\int }_B^{a{n}_{S,\lambda }^{c_{\lambda },h}}F\left(x\right)d\nu \left(x\right)$$

where λ approaches $-iq$ via the values in ${\mathbb{C}}_{+}$.

In [26,27], the authors said that real number $c_{\lambda }$ is used to explain the average (energy) value of the oscillator. If $c_{\lambda }>1$, then the oscillator increases. If $0<c_{\lambda }<1$, then the oscillator decreases. Moreover, if $c_{\lambda }<0$, then the direction of oscillator will be changed.

Let X and Y be normed spaces, and let $\mathcal{L}(X:Y)$ be the space of all bounded linear operators on X into Y. For an operator $T\in \mathcal{L}(H:H)$, the extension operator $\overline{T}$ of T on B always exists and is an element of $\mathcal{L}(B:H)$, and so its adjoint operator ${\left(\overline{T}\right)}^{*}$ is in $\mathcal{L}(H:B^{*})$. Now, let

$$\mathbb{E}=\{\overline{T}:T\in \mathcal{L}(H:H)\}.$$

Then, we have it that for each $h\in H,x\in B$ and $R\in \mathbb{E}$, we have ${(h,Rx)}^{\sim }={(R^{*}h,x)}^{\sim }.$

There are extension operators in $\mathbb{E}$ that are used in various fields. We are now explain the usefulness of operators in $\mathbb{E}$.

Remark 1. 

We give three examples of abstract Wiener space with operators [11].

(i) 

The Cameron–Martin space (infinite dimensional Hilbert space)

$$C_0^{\prime }\equiv C_0^{\prime }[0,T]=\left\{v:[0,T]\to \mathbb{R}|v\left(t\right)={\int }_0^t{z}_v\left(s\right)ds,z_v\in L_2[0,T]\right\}$$

with the norm ${\parallel v\parallel }_{C_0^{\prime }}^2={\int }_0^T{z}_v^2\left(s\right)ds$ is widely used in various fields of mathematics. Its completion with respect to the measurable norm ${\parallel v\parallel }_{C_0[0,T]}={sup}_{t\in [0,T]}\left|v\left(t\right)\right|$ is the classical Wiener space $C_0[0,T]$. In this case, the triple $(C_0^{\prime }[0,T],C_0[0,T],m_w)$ is an abstract Wiener space. Let $A_1:C_0^{\prime }\to C_0^{\prime }$ be the linear operator defined by

$$\left(A_{1}w\right)\left(t\right)={\int }_0^{t}w\left(s\right)ds.$$

(2)

Then, we see that the adjoint operator $A_1^{*}$ of $A_1$ is given by

$$A_1^{*}w\left(t\right)=w\left(T\right)t-{\int }_0^{t}w\left(s\right)ds={\int }_0^t[w\left(T\right)-w\left(s\right)]ds$$

and the linear operator $P\equiv A_1^{*}{A}_1$ is given by

$$Pw\left(t\right)={\int }_0^{T}min\{s,t\}w\left(s\right)ds.$$

One can see that P is self-adjoint on $C_0^{\prime }$ and that

$${(w_1,P{w}_2)}_{C_0^{\prime }}={(A_1{w}_1,A_1{w}_2)}_{C_0^{\prime }}={\int }_0^T{w}_1\left(s\right)w_2\left(s\right)ds$$

for all $w_1,w_2\in C_0^{\prime }$. Hence, ${(w,Pw)}_{C_0^{\prime }}\ge 0$ for all $w\in C_0^{\prime }$. We note that the orthonormal eigenfunction $\left\{e_m\right\}$ of P is given by

$$e_m\left(t\right)=\frac{\sqrt{2T}}{(m-\frac{1}{2})\pi }sin\left(\frac{(m-\frac{1}{2})\pi }{T}t\right)\equiv {\int }_0^t{\alpha }_m\left(s\right)ds$$

with corresponding eigenvalues $\left\{{\beta }_m\right\}$ given by

$${\beta }_m={\left(\frac{T}{(m-\frac{1}{2})\pi }\right)}^2.$$

Furthermore, it can be shown that $\left\{e_m\right\}$ is a basis of $C_0^{\prime }$ and so $\left\{{\alpha }_m\right\}$ is a basis of $L_2[0,T]$, and P is a trace class operator and so $A_1$ is a Hilbert–Schmidt operator on $C_0^{\prime }$. Thus, the trace of P is given by $TrP=\frac{1}{2}{T}^2={\int }_0^{T}tdt$.

(ii) 

We next consider the multiplication operator, which plays a role in physics (quantum theories); see [28]. We define a multiplication operator $A_2$ with $t\in [0,T]$ on $C_0^{\prime }$ by

$$\left(A_2\left(x\right)\right)\left(t\right)\equiv A_2\left(x\left(t\right)\right)=tx\left(t\right).$$

(3)

Then, we have $A_2\left(xy\right)=tx\left(t\right)y\left(t\right)$ and $x{A}_2\left(y\right)=x\left(t\right)ty\left(t\right).$ Moreover, one can easily check that $A_2^{*}v\left(t\right)=tv\left(t\right)$ for all $v\in C_0^{\prime }$. Note that the expected value or mean value is given by

$$E\left(x\right)\equiv {\int }_0^T{t\left|x\left(t\right)\right|}^{2}dt={\int }_0^T{A}_2{\left(\right|x|}^2)\left(t\right)dt,$$

where x is the state function of a particle in quantum mechanics and ${\int }_0^T{\left|x\left(t\right)\right|}^{2}dt$ is the probability that the particle will be founded in $[0,T]$.

(iii) 

We give another example of an abstract Wiener space. Let $H\equiv l^2$ be the space of all sequences of real numbers with $\sum _{n=1}^{\infty }{x}_n^2<\infty$. That is,

$$H\equiv l^2=\left\{\left(x_n\right):\sum _{n=1}^{\infty }{x}_n^2<\infty \right\}.$$

Its completion with respect to measurable norm $\parallel \left(x_n\right){\parallel }_0={\displaystyle \sum _{n=1}^{\infty }}\frac{1}{n^2}{x}_n^2$ is

$$B=\left\{\left(x_n\right):\sum _{n=1}^{\infty }\frac{1}{n^2}{x}_n^2<\infty \right\}.$$

Moreover, note that its dual space is

$$B^{*}=\left\{\left(x_n\right):\sum _{n=1}^{\infty }{n}^2{x}_n^2<\infty \right\}.$$

Let $R:B\to H$ be a linear operator defined by

$$R\left(\left(x_n\right)\right)=\left(\frac{1}{n}{x}_n\right).$$

Now, let $A_3{=R|}_H$. Then $A_3\in \mathcal{L}(H:H)$, $A_3$ is a self-adjoint operator and a Hilbert–Schdmit operator on H.

Using the concept of ν-lifting on abstract Wiener space, operators $A_1$ and $A_2$ can be extended on B. For more detailed study for the ν-lifting, see [2,5,7,12,16,28].

3. Existence and Relationships

In this section, we obtain the existence of a modified generalized analytic Feynman integral, and we establish some relationships for the modified generalized analytic Feynman integrals. We start this section by providing the class of functionals used in this paper. For each $w\in H$, let ${\mathsf{\Phi }}_w$ be a functional on B of the form

$${\mathsf{\Phi }}_w\left(x\right)=exp\left\{i{(w,x)}^{\sim }\right\}.$$

(4)

Let $\mathcal{E}=\left\{{\mathsf{\Phi }}_w\right|w\in H\}.$ The functionals of the form (4) are called exponential-type functionals. Let $\mathcal{A}=\mathrm{span}\mathcal{E}$ with complex coefficients. In [5,6,7], the authors showed that class $\mathcal{A}$ is a dense set in $L_2\left(B\right)$. Hence, the class $\mathcal{A}$ is an useful subject. Moreover, note that using the linearity of the integrals, we can obtain the results for functionals belonging to $\mathcal{A}$ by using the results for the functionals in $\mathcal{E}$.

In the first theorem, we show that the modified generalized analytic Feynman integral of functionals in $\mathcal{E}$ exists. Before perform this, we need the lemma described below.

Lemma 1. 

Let $S\in \mathbb{E}$ and let ${\mathsf{\Phi }}_w$ be an element of $\mathcal{E}$. Then, for $h\in H$, the generalized analytic Wiener integral ${\int }_B^{a{n}_{S,\lambda }^{c_{\lambda },h}}{\mathsf{\Phi }}_w\left(x\right)d\nu \left(x\right)$ exists for ${\mathsf{\Phi }}_w$ and is given by the following formula.

$${\int }_B^{a{n}_{S,\lambda }^{c_{\lambda },h}}{\mathsf{\Phi }}_w\left(x\right)d\nu \left(x\right)=exp\left\{-\frac{1}{2\lambda }{\left|S^{*}w\right|}_H^2+i{c}_{\lambda }{\langle w,h\rangle }_H\right\}.$$

(5)

Proof. 

For each $\lambda >0$, using Equations (1) and (4), we have

$$\begin{array}{cc}{J}_S^{c_{\lambda },h}\left({\mathsf{\Phi }}_w\right)\left(\lambda \right)& \equiv {\int }_B{\mathsf{\Phi }}_w({\lambda }^{-\frac{1}{2}}Sx+c_{\lambda }h)d\nu \left(x\right)\\ & ={\int }_{B}exp\left\{i{\lambda }^{-\frac{1}{2}}{(S^{*}w,x)}^{\sim }+i{c}_{\lambda }{\langle w,h\rangle }_H\right\}d\nu \left(x\right)\\ & =exp\left\{-\frac{1}{2\lambda }{\left|S^{*}w\right|}_H^2+i{c}_{\lambda }{\langle w,h\rangle }_H\right\}.\end{array}$$

Note that for all $\lambda >0$,

$$|J_S^{c_{\lambda },h}\left({\mathsf{\Phi }}_w\right)\left(\lambda \right)|\le \left|exp\left\{-\frac{1}{2\lambda }{\left|S^{*}w\right|}_H^2+i{c}_{\lambda }{\langle w,h\rangle }_H\right\}\right|\le |exp\left\{i{c}_{\lambda }{\langle w,h\rangle }_H\right\}|\le 1.$$

Now, for $\lambda \in {\mathbb{C}}_{+}$, let

$${\left(J_S^{c_{\lambda },h}\left({\mathsf{\Phi }}_w\right)\right)}^{*}\left(\lambda \right)=exp\left\{-\frac{1}{2\lambda }{\left|S^{*}w\right|}_H^2+i{c}_{\lambda }{\langle w,h\rangle }_H\right\}.$$

Then, ${\left(J_S^{c_{\lambda },h}\left({\mathsf{\Phi }}_w\right)\right)}^{*}\left(\lambda \right)=J_S^{c_{\lambda },h}\left({\mathsf{\Phi }}_w\right)\left(\lambda \right)$ for all $\lambda >0$. We now show that function ${\left(J_S^{c_{\lambda },h}\left({\mathsf{\Phi }}_w\right)\right)}^{*}\left(\lambda \right)$ is analytic in ${\mathbb{C}}_{+}$. In order to show this, let $\Gamma$ be any closed contour in ${\mathbb{C}}_{+}$. Then, by using Morera’s theorem, we have

$$\begin{array}{cc}{\int }_{\Gamma }{\left(J_S^{c_{\lambda },h}\left({\mathsf{\Phi }}_w\right)\right)}^{*}\left(\lambda \right)d\lambda & ={\int }_{\Gamma }exp\left\{-\frac{1}{2\lambda }{\left|S^{*}w\right|}_H^2+i{c}_{\lambda }{\langle w,h\rangle }_H\right\}d\lambda \\ & ={\int }_{\Gamma }exp\left\{-\frac{1}{2\lambda }{\left|S^{*}w\right|}_H^2+i{c}_{\lambda }{\langle w,h\rangle }_H\right\}d\lambda =0\end{array}$$

because function $exp\left\{-\frac{1}{2\lambda }{\left|S^{*}w\right|}_H^2+i{c}_{\lambda }{\langle w,h\rangle }_H\right\}$ is analytic in ${\mathbb{C}}_{+}$ as a function of $\lambda$. Thus, by the Cauchy Theorem, we complete the proof of Lemma 1 as desired. □

Theorem 1 is one of main results in this paper. In this theorem, we establish the formula for the generalized analytic Feynman integral of functionals in $\mathcal{E}$.

Theorem 1. 

Let q be a nonzero real number, and let $S,h$ and ${\mathsf{\Phi }}_w$ be the same as in Lemma 1 above. Then, the modified generalized analytic Feynman integral ${\int }_B^{an{f}_{S,q}^{c_q,h}}{\mathsf{\Phi }}_w\left(x\right)d\nu \left(x\right)$ of ${\mathsf{\Phi }}_w$ exists and is given by the following formula.

$${\int }_B^{an{f}_{S,q}^{c_q,h}}{\mathsf{\Phi }}_w\left(x\right)d\nu \left(x\right)=exp\left\{-\frac{i}{2q}{\left|S^{*}w\right|}_H^2+i{c}_q{\langle w,h\rangle }_H\right\}.$$

(6)

Proof. 

In Lemma 1 above, the existence of the generalized analytic Wiener integral was established. To complete the proof, it suffices to show that

$$\underset{\begin{array}{c}\lambda \to -iq\end{array}}{lim}{\int }_B^{a{n}_{S,\lambda }^{c_{\lambda },h}}{\mathsf{\Phi }}_w\left(x\right)d\nu \left(x\right)=exp\left\{-\frac{i}{2q}{\left|S^{*}w\right|}_H^2+i{c}_q{\langle w,h\rangle }_H\right\}.$$

Note that for a given nonzero real number q, there exists a sequence ${\left\{{\lambda }_n\right\}}_{n=1}^{\infty }$ in ${\mathbb{C}}_{+}$ so that ${\lambda }_n\to -iq$ as $n\to \infty$. Hence, for all nonzero real number q,

$$\begin{array}{cc}\underset{\lambda \to -iq}{lim}{\int }_B^{a{n}_{S,\lambda }^{c_{\lambda },h}}{\mathsf{\Phi }}_w\left(x\right)d\nu \left(x\right)& =\underset{n\to \infty }{lim}exp\left\{-\frac{1}{2{\lambda }_n}{\left|S^{*}w\right|}_H^2+i{c}_{\lambda }{\langle w,h\rangle }_H\right\}\\ & =exp\left\{-\frac{i}{2q}{\left|S^{*}w\right|}_H^2+i{c}_q{\langle w,h\rangle }_H\right\}\end{array}$$

which establishes Equation (6) as desired. Furthermore,

$$\left|{\int }_B^{an{f}_{S,q}^{c_q,h}}{\mathsf{\Phi }}_w\left(x\right)d\nu \left(x\right)\right|\le exp\left\{|c_q{\left|\right|w|}_H^2{\left|h\right|}_H^2\right\}<\infty .$$

Hence, we complete the proof of Theorem 1. □

In our next theorem, we obtain the Fubini theorem and the composition formula with respect to the modified generalized analytic Feynman integrals.

Theorem 2. 

Let ${\mathsf{\Phi }}_w$ be an element of $\mathcal{E}$ and let $S_1,S_2\in \mathbb{E}$. Let $h_1$ and $h_2$ be elements of H. Then, for all nonzero real numbers $q_1$ and $q_2$ with $q_1+q_2\ne 0$, we have

$$\begin{array}{c}{\int }_B^{an{f}_{S_1,q_1}^{c_{q_1},h_1}}{\int }_B^{an{f}_{S_2,q_2}^{c_{q_2},h_2}}{\mathsf{\Phi }}_w(x_1+x_2)d\nu \left(x_1\right)d\nu \left(x_2\right)\\ ={\int }_B^{an{f}_{S_3,q_3}^{c_{q_3},h_3}}{\mathsf{\Phi }}_w\left(x\right)d\nu \left(x\right)\\ ={\int }_B^{an{f}_{S_2,q_2}^{c_{q_2},h_2}}{\int }_B^{an{f}_{S_1,q_1}^{c_{q_1},h_1}}{\mathsf{\Phi }}_w(x_1+x_2)d\nu \left(x_2\right)d\nu \left(x_1\right)\end{array}$$

(7)

where

$$\frac{1}{q_1}|S_1^{*}{w|}_H^2+\frac{1}{q_2}|S_2^{*}{w|}_H^2=\frac{1}{q_3}{\left|S_3^{*}w\right|}_H^2\mathrm{and}{c}_{q_1}{h}_1+c_{q_2}{h}_2=c_{q_3}{h}_3.$$

(8)

In particular, for $S\in \mathcal{L}$, we have

$${\int }_B^{an{f}_{S,q_1}^{c_{q_1},h_1}}{\int }_B^{an{f}_{S,q_2}^{c_{q_2},h_2}}{\mathsf{\Phi }}_w(x_1+x_2)d\nu \left(x_1\right)d\nu \left(x_2\right)={\int }_B^{an{f}_{S,q_3}^{c_{q_3},h_3}}{\mathsf{\Phi }}_w\left(x\right)d\nu \left(x\right)$$

(9)

where $q_3=\frac{q_1{q}_2}{q_1+q_2}$.

Proof. 

First, for all ${\lambda }_1>0$ and ${\lambda }_2>0$, we have

$$\begin{array}{c}{\int }_B{\int }_B{\mathsf{\Phi }}_w({\lambda }_1^{-\frac{1}{2}}{S}_1{x}_1+{\lambda }_2^{-\frac{1}{2}}{S}_2{x}_2+c_{{\lambda }_1}{h}_1+c_{{\lambda }_2}{h}_2)d\nu \left(x_1\right)d\nu \left(x_2\right)\\ ={\int }_B{\int }_{B}exp\left\{i\sum _{j=1}^2{\lambda }_j^{-\frac{1}{2}}{(S_j^{*}w,x_j)}^{\sim }+i\sum _{j=1}^2{c}_{{\lambda }_j}{\langle w,h_j\rangle }_H\right\}d\nu \left(x_1\right)d\nu \left(x_2\right)\\ =exp\left\{-\sum _{j=1}^2\frac{1}{2{\lambda }_j}{\left|S_j^{*}w\right|}_H^2+i\sum _{j=1}^2{c}_{{\lambda }_j}{\langle w,h_j\rangle }_H\right\}\\ =exp\left\{-\sum _{j=1}^2\frac{1}{2{\lambda }_j}{\left|S_j^{*}w\right|}_H^2+i{\langle w,c_{{\lambda }_1}{h}_1+c_{{\lambda }_2}{h}_2\rangle }_H\right\}\end{array}$$

and from Equation (1) above, we have

$${\int }_{B}F({\lambda }_3^{-\frac{1}{2}}{S}_{3}x+c_{{\lambda }_3}{h}_3)d\nu \left(x\right)=exp\left\{-\frac{1}{2{\lambda }_3}{\left|S_3^{*}w\right|}_H^2+i{c}_{{\lambda }_3}{\langle w,h_3\rangle }_H\right\}.$$

These equations mean that

$${\int }_B^{a{n}_{S_1,{\lambda }_1}^{c_{{\lambda }_1},h_1}}{\int }_B^{a{n}_{S_2,{\lambda }_2}^{c_{{\lambda }_2},h_2}}{\mathsf{\Phi }}_w(x_1+x_2)d\nu \left(x_1\right)d\nu \left(x_2\right)={\int }_B^{a{n}_{S_3,{\lambda }_3}^{c_{{\lambda }_3},h_3}}{\mathsf{\Phi }}_w\left(x\right)d\nu \left(x\right)$$

if the following is the case.

$$\frac{1}{{\lambda }_1}|S_1^{*}{w|}_H^2+\frac{1}{{\lambda }_2}|S_2^{*}{w|}_H^2=\frac{1}{{\lambda }_3}{\left|S_3^{*}w\right|}_H^2\mathrm{and}{c}_{{\lambda }_1}{h}_1+c_{{\lambda }_2}{h}_2=c_{{\lambda }_3}{h}_3.$$

(10)

Next, they can be analytically continued in ${\lambda }_1,{\lambda }_2$, and ${\lambda }_3$ on ${\mathbb{C}}_{+}$ and so by letting ${\lambda }_j\to -i{q}_j$, $j=1,2,3$, we can establish the first equality in Equation (7) as desired. Furthermore, the second equality immediately follows from the symmetric property for abstract Wiener integrals. Moreover, if $S_1=S_2=S_3=S$, then Equation (9) is obtained easily from Equation (7). Hence, we have the desired results. □

We next shall provide some modified generalized analytic Feynman integration formulas via the translation theorem on abstract Wiener spaces. Equation (11) below is called the basic translation theorem on abstract Wiener spaces; see [4,5,6,7,12].

Lemma 2. 

(Translation theorem)Let F be an integrable functional on B and let $x_0\in H$. Then,

$${\int }_{B}F(x+x_0)d\nu \left(x\right)=exp\left\{-\frac{1}{2}{\left|x_0\right|}_H^2\right\}{\int }_{B}F\left(x\right)exp\left\{{(x_0,x)}^{\sim }\right\}d\nu \left(x\right).$$

(11)

We establish two formulas with respect to the modified generalized analytic Feynman integral via the translation theorem. The first result is a connection between the modified generalized analytic Feynman integral and the modified generalized analytic Feynman integral with the exponential weight.

Theorem 3. 

Let ${\mathsf{\Phi }}_w,S,q$, and h be the same as in Theorem 1 above. Then, we have

$$\begin{array}{cc}{\int }_B^{an{f}_{S,q}^{c_q,h}}{\mathsf{\Phi }}_w\left(x\right)d\nu \left(x\right)& =exp\left\{\frac{iq{c}_q^2}{2}|S^{*}{h|}_H^2+iq{c}_q^2{\left|h\right|}_H^2-i{c}_q{\langle w,h\rangle }_H\right\}\\ & \times {\int }_B^{an{f}_{S,q}^{c_q,h}}{\mathsf{\Phi }}_w\left(x\right)exp\{-iq{c}_q{(h,x)}^{\sim }\}d\nu \left(x\right).\end{array}$$

(12)

Proof. 

First, for $\lambda >0$, we note that

$$J_S^{c_{\lambda },h}\left({\mathsf{\Phi }}_w\right)\left(\lambda \right)={\int }_B{\mathsf{\Phi }}_w({\lambda }^{-\frac{1}{2}}Sx+c_{\lambda }h)d\nu \left(x\right).$$

Now, let ${\left({\mathsf{\Phi }}_w\right)}_{\lambda }\left(x\right)={\mathsf{\Phi }}_w\left({\lambda }^{-\frac{1}{2}}x\right)$ and let ${\mathsf{\Phi }}_w^S\left(x\right)={\mathsf{\Phi }}_w\left(Sx\right)$. Then, we have the following expression.

$$\begin{array}{cc}{\mathsf{\Phi }}_w({\lambda }^{-\frac{1}{2}}Sx+c_{\lambda }h)& ={\mathsf{\Phi }}_w\left({\lambda }^{-\frac{1}{2}}(Sx+{\lambda }^{\frac{1}{2}}{c}_{\lambda }h)\right)={\left({\mathsf{\Phi }}_w\right)}_{\lambda }(Sx+{\lambda }^{\frac{1}{2}}{c}_{\lambda }h)\\ & ={\left({\mathsf{\Phi }}_w\right)}_{\lambda }\left(S(x+{\lambda }^{\frac{1}{2}}{c}_{\lambda }{S}^{*}h)\right)={\left({\mathsf{\Phi }}_w\right)}_{\lambda }^S(x+{\lambda }^{\frac{1}{2}}{c}_{\lambda }{S}^{*}h).\end{array}$$

Since $h\in H$ and $S\in \mathbb{E}$, $S^{*}h\in B^{*}\subset H$; thus, we apply Equation (11) to functional ${\left({\mathsf{\Phi }}_w\right)}_{\lambda }^S$ instead of F with $x_0={\lambda }^{\frac{1}{2}}{c}_{\lambda }{S}^{*}h$, and we have

$$\begin{array}{c}{\int }_B{\mathsf{\Phi }}_w({\lambda }^{-\frac{1}{2}}Sx+c_{\lambda }h)d\nu \left(x\right)\\ =exp\left\{-\frac{\lambda c_{\lambda }^2}{2}{\left|S^{*}h\right|}_H^2\right\}{\int }_B{\left({\mathsf{\Phi }}_w\right)}_{\lambda }^S\left(x\right)exp\left\{{\lambda }^{\frac{1}{2}}{c}_{\lambda }{(S^{*}h,x)}^{\sim }\right\}d\nu \left(x\right)\\ =exp\left\{-\frac{\lambda c_{\lambda }^2}{2}{\left|S^{*}h\right|}_H^2\right\}{\int }_B{\mathsf{\Phi }}_w\left({\lambda }^{-\frac{1}{2}}Sx\right)exp\left\{\lambda c_{\lambda }{(h,{\lambda }^{-\frac{1}{2}}Sx)}^{\sim }\right\}d\nu \left(x\right).\end{array}$$

We next note that

$$\begin{array}{cc}{\mathsf{\Phi }}_w\left({\lambda }^{-\frac{1}{2}}Sx\right)& =exp\{i{\lambda }^{-\frac{1}{2}}{(w,Sx)}^{\sim }+i{c}_{\lambda }{\langle w,h\rangle }_H-i{c}_{\lambda }{\langle w,h\rangle }_H\}\\ & =exp\{-i{c}_{\lambda }{\langle w,h\rangle }_H\}{\mathsf{\Phi }}_w({\lambda }^{-\frac{1}{2}}Sx+c_{\lambda }h)\end{array}$$

and hence

$$\begin{array}{c}{\int }_B{\mathsf{\Phi }}_w({\lambda }^{-\frac{1}{2}}Sx+c_{\lambda }h)d\nu \left(x\right)\\ =exp\left\{-\frac{\lambda c_{\lambda }^2}{2}|S^{*}{h|}_H^2-\lambda c_{\lambda }^2{\left|h\right|}_H^2-i{c}_{\lambda }{\langle w,h\rangle }_H\right\}\\ \times {\int }_B{\mathsf{\Phi }}_w({\lambda }^{-\frac{1}{2}}Sx+c_{\lambda }h)exp\left\{\lambda c_{\lambda }{(h,{\lambda }^{-\frac{1}{2}}Sx+c_{\lambda }h)}^{\sim }\right\}d\nu \left(x\right).\end{array}$$

It can be analytically continued in $\lambda \in {\mathbb{C}}_{+}$ and by letting $\lambda \to -iq$, we have

$$\begin{array}{cc}{\int }_B^{an{f}_{S,q}^{c_q,h}}{\mathsf{\Phi }}_w\left(x\right)d\nu \left(x\right)& =exp\left\{\frac{iq{c}_q^2}{2}|S^{*}{h|}_H^2+iq{c}_q^2{\left|h\right|}_H^2-i{c}_q{\langle w,h\rangle }_H\right\}\\ & \times {\int }_B^{an{f}_{S,q}^{c_q,h}}{\mathsf{\Phi }}_w\left(x\right)exp\{-iq{c}_q{(h,x)}^{\sim }\}d\nu \left(x\right).\end{array}$$

Hence, we have the desired results. □

The second result is a connection between the modified generalized analytic Feynman integral and the ordinary analytic Feynman integral.

Theorem 4. 

Let ${\mathsf{\Phi }}_w,S,q$, and h be the same as in Theorem 1 above. Assume that $S{S}^{*}=I$ on H. Then, we have

$$\begin{array}{c}{\int }_B^{an{f}_{S,q}^{c_q,h}}{\mathsf{\Phi }}_w\left(x\right)d\nu \left(x\right)\\ =exp\left\{\frac{iq{c}_q^2}{2}{\left|S^{*}h\right|}_H^2\right\}{\int }_B^{an{f}_q}{\mathsf{\Phi }}_w\left(Sx\right)exp\{-iq{c}_q{(h,Sx)}^{\sim }\}d\nu \left(x\right).\end{array}$$

(13)

Proof. 

For $\lambda >0$, let the following be the case.

$$J_S^{c_{\lambda },h}\left(\lambda \right)={\int }_B{\mathsf{\Phi }}_w({\lambda }^{-\frac{1}{2}}Sx+c_{\lambda }h)d\nu \left(x\right).$$

Using the similar method in the proof of Theorem 3, we apply Equation (11) to functional ${\left({\mathsf{\Phi }}_w\right)}_{\lambda }^S$ instead of F with $x_0={\lambda }^{\frac{1}{2}}{c}_{\lambda }{S}^{*}h$, and we have

$$\begin{array}{c}{\int }_B{\mathsf{\Phi }}_w({\lambda }^{-\frac{1}{2}}Sx+c_{\lambda }h)d\nu \left(x\right)\\ =exp\left\{-\frac{\lambda c_{\lambda }^2}{2}{\left|S^{*}h\right|}_H^2\right\}{\int }_B{\left({\mathsf{\Phi }}_w\right)}_{\lambda }^S\left(x\right)exp\left\{{\lambda }^{\frac{1}{2}}{c}_{\lambda }{(S^{*}h,x)}^{\sim }\right\}d\nu \left(x\right)\\ =exp\left\{-\frac{\lambda c_{\lambda }^2}{2}{\left|S^{*}h\right|}_H^2\right\}{\int }_B{\mathsf{\Phi }}_w\left({\lambda }^{-\frac{1}{2}}Sx\right)exp\left\{\lambda c_{\lambda }{(h,{\lambda }^{-\frac{1}{2}}Sx)}^{\sim }\right\}d\nu \left(x\right).\end{array}$$

It can be analytically continued in $\lambda \in {\mathbb{C}}_{+}$ and letting $\lambda \to -iq$, we have

$$\begin{array}{c}{\int }_B^{an{f}_{S,q}^{c_q,h}}{\mathsf{\Phi }}_w\left(x\right)d\nu \left(x\right)\\ =exp\left\{\frac{iq{c}_q^2}{2}{\left|S^{*}h\right|}_H^2\right\}{\int }_B^{an{f}_q}{\mathsf{\Phi }}_w\left(Sx\right)exp\{-iq{c}_q{(h,Sx)}^{\sim }\}d\nu \left(x\right),\end{array}$$

which completes the proof of Theorem 4 as desired. □

We finish this section by providing the (modified) generalized analytic Feynman integration formulas for the exponential functional.

Corollary 1. 

Let ${\mathsf{\Phi }}_w\left(x\right)=1$ for all $x\in B$ in Theorem 4. Then, we have

$${\int }_B^{an{f}_q}exp\{-iq{c}_q{(h,Sx)}^{\sim }\}d\nu \left(x\right)=exp\left\{-\frac{iq{c}_q^2}{2}{\left|S^{*}h\right|}_H^2\right\}$$

(14)

and

$$\begin{array}{cc}{\int }_B^{an{f}_{S,q}^{c_q,h}}& exp\{-iq{c}_q{(h,x)}^{\sim }\}d\nu \left(x\right)\\ & =exp\left\{-\frac{iq{c}_q^2}{2}|S^{*}{h|}_H^2-iq{c}_q^2{\left|h\right|}_H^2+i{c}_q{\langle w,h\rangle }_H\right\}.\end{array}$$

(15)

Comparing Equations (14) and (15), we can find a difference between the modified generalized analytic Feynman integrals and the ordinary analytic Feynman integrals.

4. Cameron–Storvick Theorems

In this section, we establish the Cameron–Storvick theorems for the modified generalized analytic Feynman integrals. Before we perform this, we need the concept of a generalized first variation for functionals on B.

Definition 2. 

Let S be an element of $\mathbb{E}$ and let F be a functional on B. Then, the generalized first variation ${\delta }^{S}F\left(x\right|u)$ of F is defined by the following formula

$${\delta }^{S}F\left(x\right|u)=\frac{\partial }{\partial \alpha }F(x+\alpha Su){|}_{\alpha =0},$$

(16)

for $x,u\in B$ if it exists.

In Lemma 3 below, we establish the existence of generalized first variation for the functionals in $\mathcal{E}$.

Lemma 3. 

Let S be an element of $\mathbb{E}$ and let ${\mathsf{\Phi }}_w$ be an element of $\mathcal{E}$. Then, for $u\in H$, the generalized first variation ${\delta }^S{\mathsf{\Phi }}_w\left(x\right|u)$ of ${\mathsf{\Phi }}_w$ exists and is given by the following formula.

$${\delta }^S{\mathsf{\Phi }}_w\left(x\right|u)=i{\langle S^{*}w,u\rangle }_{H}exp\left\{i{(w,x)}^{\sim }\right\}.$$

(17)

Proof. 

Using Equations (1) and (16), we have

$$\begin{array}{cc}{\delta }^S{\mathsf{\Phi }}_w\left(x\right|u)& =\frac{\partial }{\partial \alpha }exp\left\{i{(w,x)}^{\sim }+i\alpha \langle S^{*}w,u\rangle \right\}{|}_{\alpha =0}\\ & =i{\langle S^{*}w,u\rangle }_{H}exp\left\{i{(w,x)}^{\sim }\right\}.\end{array}$$

Now, using Holder’s inequality, we have

$$|{\delta }^S{\mathsf{\Phi }}_w\left(x\right|u)|\le |{\langle S^{*}w,u\rangle }_H|\le |S^{*}{w|}_H{\left|u\right|}_H\le \parallel S^{*}{\parallel }_{op}{\left|w\right|}_H{\left|u\right|}_H<\infty$$

where ${\parallel T\parallel }_{op}$ denotes the operator norm of an operator T; hence, we complete the proof of Lemma 3. □

In Theorem 5, we provide a formula for the modified generalized analytic Feynman integral involving the generalized first variation.

Theorem 5. 

Let $S_1$ and $S_2$ be the elements of $\mathbb{E}$ and let ${\mathsf{\Phi }}_w$ be an element of $\mathcal{E}$. Then, for $u\in H$, the modified generalized analytic Feynman integral ${\int }_B^{an{f}_{S_1,q}^{c_q,h}}{\delta }^{S_2}{\mathsf{\Phi }}_w\left(x\right|u)d\nu \left(x\right)$ involving the generalized first variation exists, and it is given by the following formula.

$${\int }_B^{an{f}_{S_1,q}^{c_q,h}}{\delta }^{S_2}{\mathsf{\Phi }}_w\left(x\right|u)d\nu \left(x\right)=i{\langle S_2^{*}w,u\rangle }_{H}exp\left\{\frac{i}{2q}{\left|S_1^{*}w\right|}_H^2+i{c}_q{\langle w,u\rangle }_H\right\}.$$

(18)

Proof. 

We proved that the generalized first variation of ${\delta }^{S_2}{\mathsf{\Phi }}_w\left(x\right|u)$ exists and is given by the following formula.

$${\delta }^{S_2}{\mathsf{\Phi }}_w\left(x\right|u)=i{\langle S_2^{*}w,u\rangle }_{H}exp\left\{i{(w,x)}^{\sim }\right\}.$$

By using Equations (6) and (17), Equation (18) is obtained as desired. Hence, we complete the proof of Theorem 5. □

The Cameron–Storvick theorem shows that the Wiener integrals involving the first variation can be expressed by the ordinary forms without the concept of the first variation. For this reason, it is also called the integration by the parts formula. Numerous constructions and theories regarding the Cameron–Storvick theorem have been studied and applied in many papers [4,5,14,15,17].

We shall establish a more generalized Cameron–Storvick theorem with respect to the our generalized analytic Feynman integral with the generalized first variation. Using Equation (11), we establish a translation theorem to obtain the generalized Cameron–Storvick theorem, which was established in ([29], Theorem 4.2).

Lemma 4. 

(Translation theorem with operators.)Let $S_1$ and $S_2$ be elements of $\mathbb{E}$ with $S_1$ being the unitary operator on H. Let F be an integrable functional on B and let $x_0\in H$. Then, we have the following.

$$\begin{array}{cc}{\int }_{B}F(S_{1}x+S_2{x}_0)d\nu \left(x\right)& =exp\left\{-\frac{1}{2}{\left|S_1^{*}{S}_2{x}_0\right|}_H^2\right\}\\ & \times {\int }_{B}F\left(S_{1}x\right)exp\left\{{(S_1^{*}{S}_2{x}_0,x)}^{\sim }\right\}d\nu \left(x\right).\end{array}$$

(19)

Equation (20) below is called the generalized Cameron–Storvick theorem.

Theorem 6. 

Let $S_1,S_2,{\mathsf{\Phi }}_w,h$, and u be the same as in Theorem 5 above. Then, the following is the case.

$$\begin{array}{cc}{\int }_B^{an{f}_{S_1,q}^{c_q,h}}{\delta }^{S_2}{\mathsf{\Phi }}_w\left(x\right|u)d\nu \left(x\right)& =-iq{\int }_B^{an{f}_{S_1,q}^{c_q,h}}{(S_1^{*}{S}_{2}w,x)}^{\sim }{\mathsf{\Phi }}_w\left(x\right)d\nu \left(x\right)\\ & +iq{c}_q{\langle S_1^{*}{S}_{2}w,h\rangle }_H{\int }_B^{an{f}_{S_1,q}^{c_q,h}}{\mathsf{\Phi }}_w\left(x\right)d\nu \left(x\right).\end{array}$$

(20)

Proof. 

The existence of Equation (20) was established in Theorem 5 above. We retain the position that the equality holds in Equation (20). For $\lambda >0$, we have

$${\int }_B{\delta }^{S_2}{\mathsf{\Phi }}_w({\lambda }^{-\frac{1}{2}}{S}_{1}x+c_{\lambda }h|u)d\nu \left(x\right)={\int }_B[\frac{\partial }{\partial \alpha }{\mathsf{\Phi }}_w({\lambda }^{-\frac{1}{2}}{S}_{1}x+{\alpha }_1{S}_{2}u+c_{\lambda }h)\left|{}_{\alpha =0}\right]d\nu \left(x\right).$$

In order to apply Equation (19), let ${\left({\mathsf{\Phi }}_w\right)}_h\left(x\right)={\mathsf{\Phi }}_w(x+c_{\lambda }h)$ and ${\left({\mathsf{\Phi }}_w\right)}_h^{\lambda }={\left({\mathsf{\Phi }}_w\right)}_h\left({\lambda }^{-\frac{1}{2}}x\right).$

Then, ${\mathsf{\Phi }}_w({\lambda }^{-\frac{1}{2}}{S}_{1}x+{\alpha }_1{S}_{2}u+c_{\lambda }h)={\left({\mathsf{\Phi }}_w\right)}_h^{\lambda }(S_{1}x+\alpha {\lambda }^{\frac{1}{2}}{S}_{2}u)$. Thus, for $x_0\equiv \alpha {\lambda }^{\frac{1}{2}}u$,

$$\begin{array}{c}{\int }_B{\delta }^{S_2}{\mathsf{\Phi }}_w({\lambda }^{-\frac{1}{2}}{S}_{1}x+c_{\lambda }h|u)d\nu \left(x\right)\\ =\frac{\partial }{\partial \alpha }[exp\left\{-\frac{\lambda {\alpha }^2}{2}{\left|S_1^{*}{S}_{2}u\right|}_H^2\right\}\\ \times {\int }_B{\left({\mathsf{\Phi }}_w\right)}_h\left({\lambda }^{-\frac{1}{2}}{S}_{1}x\right)exp\left\{{\lambda }^{\frac{1}{2}}\alpha {(S_1^{*}{S}_{2}w,x)}^{\sim }\right\}d\nu \left(x\right)]{|}_{\alpha =0}\\ ={\lambda }^{\frac{1}{2}}{\int }_B{(S_1^{*}{S}_{2}w,x)}^{\sim }{\mathsf{\Phi }}_w({\lambda }^{-\frac{1}{2}}{S}_{1}x+c_{\lambda }h)d\nu \left(x\right)\\ =\lambda {\int }_B{(S_1^{*}{S}_{2}w,{\lambda }^{-\frac{1}{2}}x+c_{\lambda }h)}^{\sim }{\mathsf{\Phi }}_w({\lambda }^{-\frac{1}{2}}{S}_{1}x+c_{\lambda }h)d\nu \left(x\right)\\ -\lambda c_{\lambda }{\langle S_1^{*}{S}_{2}w,h\rangle }_H{\int }_B{\mathsf{\Phi }}_w({\lambda }^{-\frac{1}{2}}{S}_{1}x+c_{\lambda }h)d\nu \left(x\right).\end{array}$$

By similar methods as in the proof of Theorem 1, we obtain it by analytical continuation into ${\mathbb{C}}_{+}$, and by letting $\lambda \to -iq$, we have

$$\begin{array}{cc}{\int }_B^{an{f}_{S_1,q}^{c_q,h}}{\delta }^{S_2}{\mathsf{\Phi }}_w\left(x\right|u)d\nu \left(x\right)& =-iq{\int }_B^{an{f}_{S_1,q}^{c_q,h}}{(S_1^{*}{S}_{2}w,x)}^{\sim }{\mathsf{\Phi }}_w\left(x\right)d\nu \left(x\right)\\ & +iq{c}_q{\langle S_1^{*}{S}_{2}w,h\rangle }_H{\int }_B^{an{f}_{S_1,q}^{c_q,h}}{\mathsf{\Phi }}_w\left(x\right)d\nu \left(x\right).\end{array}$$

Hence, we have the desired results. □

By taking Theorems 3 and 4 together with Theorem 3, the following corollary below is obtained easily.

Corollary 2. 

Let $S_1,S_2,{\mathsf{\Phi }}_w,h$, and u be the same as in Theorem 5 above. Then, we have the following.

$$\begin{array}{cc}{\int }_B^{an{f}_{S_1,q}^{c_q,h}}{\delta }^{S_2}{\mathsf{\Phi }}_w\left(x\right|u)d\nu \left(x\right)& =exp\left\{\frac{iq{c}_q^2}{2}|S_1^{*}{h|}_H^2+iq{c}_q^2{\left|h\right|}_H^2-i{c}_q{(w,h)}_H\right\}\\ & \times i{\langle S_2^{*}w,u\rangle }_H{\int }_B^{an{f}_{S_1,q}^{c_q,h}}{\mathsf{\Phi }}_w\left(x\right)exp\{-iq{c}_q{(h,x)}^{\sim }\}d\nu \left(x\right)\\ & =i{\langle S_2^{*}w,u\rangle }_{H}exp\left\{\frac{iq{c}_q^2}{2}{\left|S_1^{*}h\right|}_H^2\right\}\\ & \times {\int }_B^{an{f}_q}{\mathsf{\Phi }}_w\left(S_{1}x\right)exp\{-iq{c}_q{(h,S_{1}x)}^{\sim }\}d\nu \left(x\right).\end{array}$$

(21)

Furthermore if ${\mathsf{\Phi }}_w\left(x\right)=1$ for all $x\in B$ in Theorem 4, then ${\int }_B^{an{f}_{S_1,q}^{c_q,h}}{\delta }^{S_2}{\mathsf{\Phi }}_w\left(x\right|u)d\nu \left(x\right)=0$, and so we have

$${\int }_B^{an{f}_{S_1,q}^{c_q,h}}{(S_1^{*}{S}_{2}w,x)}^{\sim }d\nu \left(x\right)=c_q{\langle S_1^{*}{S}_{2}w,h\rangle }_H$$

and

$${\int }_B^{an{f}_q}exp\{-iq{c}_q{(h,Sx)}^{\sim }\}d\nu \left(x\right)=exp\left\{-\frac{iq{c}_q^2}{2}{\left|S^{*}h\right|}_H^2\right\}.$$

5. Applications

We finish this paper by providing some applications of our results and formulas.

The modified generalized analytic Feynman integral

$${\int }_B^{an{f}_{S,q}^{c_q,h}}{\mathsf{\Phi }}_w\left(x\right)exp\{-iq{c}_q{(h,x)}^{\sim }\}d\nu \left(x\right)$$

(22)

is not easy to calculate because the integrand involves an exponential weight. As such, a calculation of the following Wiener integral

$${\int }_{B}exp\{{(w_1,x)}^{\sim }+{(w_2,x)}^{\sim }\}d\nu \left(x\right)$$

is not easy unless $w_1$ and $w_2$ are orthogonal. However, from Equations (6) and (13), we have

$$\begin{array}{c}{\int }_B^{an{f}_{S,q}^{c_q,h}}{\mathsf{\Phi }}_w\left(x\right)exp\{-iq{c}_q{(h,x)}^{\sim }\}d\nu \left(x\right)\\ =exp\left\{-\frac{iq{c}_q^2}{2}|S^{*}{h|}_H^2-iq{c}_q^2{\left|h\right|}_H^2+i{c}_q{\langle w,h\rangle }_H\right\}{\int }_B^{an{f}_{S,q}^{c_q,h}}{\mathsf{\Phi }}_w\left(x\right)d\nu \left(x\right)\\ =exp\left\{-\frac{i}{2q}|S^{*}{w|}_H^2-\frac{iq{c}_q^2}{2}|S^{*}{h|}_H^2-iq{c}_q^2{\left|h\right|}_H^2+2i{c}_q{\langle w,h\rangle }_H\right\}.\end{array}$$

(23)

Equation (23) tells us that we can calculate a modified generalized analytic Feynman integral of form (22). We shall explain this usefulness via two examples.

Case 1: Multiplication operator on $({\mathbf{C}}_{\mathbf{0}}^{\prime }[\mathbf{0},\mathbf{T}],{\mathbf{C}}_{\mathbf{0}}[\mathbf{0},\mathbf{T}],\mathbf{\nu })$.

For the multiplication operator, $A\equiv A_2$ is the same as in Remark 1. Let S be the extension operator of A. Let $w_0\left(t\right)={\int }_0^{t}cos\left(s\right)ds=sin\left(t\right)$ and let $h\left(t\right)={\int }_0^{t}1ds=t$. Then, $w_0$ and h in $C_0^{\prime }[0,T]$. Then, we have

$$\begin{array}{c}|S^{*}{w}_0{|}_{C_0^{\prime }}^2=\frac{t^{2}T}{2}+\frac{t^2}{4}sin2T,\\ |S^{*}{h|}_{C_0^{\prime }}^2=t^{2}T,\\ {(w_0,h)}_{C_0^{\prime }}=sinT,\\ {\left|h\right|}_{C_0^{\prime }}^2=T.\end{array}$$

Using Equation (6), we have

$${\int }_B^{an{f}_{S,q}^{c_q,h}}{\mathsf{\Phi }}_{w_0}\left(x\right)d\nu \left(x\right)=exp\left\{-\frac{i}{2q}\left(\frac{t^{2}T}{2}+\frac{t^2}{4}sin2T\right)+i{c}_{q}sinT\right\}.$$

Furthermore, using Equation (23), we have

$$\begin{array}{c}{\int }_B^{an{f}_{S,q}^{c_q,h}}{\mathsf{\Phi }}_{w_0}\left(Sx\right)exp\{-iq{c}_q{(h,Sx)}^{\sim }\}d\nu \left(x\right)\\ =exp\left\{-\frac{i}{2q}\left(\frac{t^{2}T}{2}+\frac{t^2}{4}sin2T\right)-\frac{iq{c}_q^2{t}^{2}T}{2}+i{c}_q(2sinT-qT)\right\}.\end{array}$$

Case 2: Sequential operator on $({\mathbf{l}}^{\mathbf{2}},\mathbf{B},\mathbf{\nu })$.

For a bounded linear operator $A\equiv A_3$, let $w_0\equiv x_n=\frac{1}{n^2}$ and let $h\equiv h_n=\frac{1}{n^2}$. Then, ${\displaystyle \sum _{n=1}^{\infty }}{n}^2{x}_n^2=\frac{{\pi }^2}{6}<\infty$ and so $w\in B^{*}\subset H\equiv l^2\subset B$. For each $k=2,3,\dots$, let $a_k={\displaystyle \sum _{n=1}^{\infty }}\frac{1}{n^k}$. Then, we have

$$\begin{array}{c}|R^{*}{w}_0{|}_{l^2}^2={\displaystyle \sum _{n=1}^{\infty }}\frac{1}{n^6}=a_6,\\ |R^{*}{h|}_{l^2}^2={\displaystyle \sum _{n=1}^{\infty }}\frac{1}{n^6}=a_6,\\ {(w_0,h)}_{l^2}={\displaystyle \sum _{n=1}^{\infty }}\frac{1}{n^4}=a_4,\\ {\left|h\right|}_{l^2}^2={\displaystyle \sum _{n=1}^{\infty }}\frac{1}{n^4}=a_4.\end{array}$$

Using Equation (6), we have

$${\int }_B^{an{f}_{S,q}^{c_q,h}}{\mathsf{\Phi }}_{w_0}\left(x\right)d\nu \left(x\right)=exp\left\{-\frac{i{a}_6}{2q}+i{c}_q{a}_4\right\}.$$

Furthermore, using Equation (23), we have

$${\int }_B^{an{f}_{S,q}^{c_q,h}}{\mathsf{\Phi }}_{w_0}\left(Sx\right)exp\{-iq{c}_q{(h,Sx)}^{\sim }\}d\nu \left(x\right)=exp\left\{-\frac{i{a}_6}{2q}-\frac{iq{c}_q^2{a}_6}{2}+i{c}_q{a}_4(2-q)\right\}.$$